Sealed Solution

1177 /www/nrich/html/content/03/06/penta4/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>A set of ten cards, each showing one of the digits from $0$ to $9$, is divided up between five envelopes so that there are two cards in each envelope. The sum of the two numbers inside it is written on each envelope:</p> <p class="c1"><mdo:image alt="five envelopes with the numbers 7, 8, 13, 14 and 3 written on them (one number per envelope)" height="35" src="envelopes.gif" width="288"></mdo:image></p> <p class="c2">What numbers could be inside the $8$ envelope?</p> <p class="acknowledgement c2" style="font-style: italic;">Thank you to Alan Parr for allowing us to adapt one of his problems.</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><span class="editorial">We had over 100 solutions for this challenge.<br></br> From Heage school we had solutions from Miles, Eleanor, Ellie, Erin, Olivia, Brogan, Ruby and Phoebe.<br></br> From Riverside International School of Prague in the Czech Republic we had solutions from Sabrina, George, Tereza, Ieuan, Catalina, Jewoo, Daniel and Michelle.<br></br> From Thameside primary School we had solutions from Devon, Jade, Rebekah, Harry, Jordan, Chelsea.<br></br> From St. Helen's C of E School solutions came from Jay, Neve, Lily, Alice, Saba, Sam, Jamie, Thomas, Will, Ollie, Aaron, Oliver, Jack, Katie, Tilly, Holly, Matthew, Caitlin, Esme, Oliver and Courtney.<br></br> From Brewers Hill School Paige, Brandon, Paul, Daniel, Shannon, Leigha, Kai, Maia-storme, Max, John, Lucy and Jacob sent in solutions.<br></br> From Shebbear Community School we had solutions from Rebecca, Maddie, Megs, Samual, Lewis, Grace, Emma, Ellie, Paige, Megan, Jade, Kelsy and Keeley.<br></br> From Clatford C of E Primary School Finn, Hugo, Sydney, Alice, Tabby sent in solutions.<br></br> From Paddox Primary School we had solutions from Arjan, Lucy, Stephanie, Claire, Charlie, Jess and Mia. <br></br> Lewis from Oatlands School sent in a very interesting solution in a Word format unfortunately it was too pale to satisfactorily transfer for us.</span><br></br> <span class="editorial">Other solutions came in from Ruth & Martha, Rober, Daniel, Maddison, Bethany, Alex, Oliver, Claire, Mlee, George, Gokhan, Shay Adesh, Seval, Erica, Barney, Jahelm, Ayaan, Jack, Rosa, Wilf, Finn, Ralph & Niamh, Miss Blake's Mathematical Team of clever girls, Kieran, Thomas, Miyazur, Olti, Year 6, Anna, Isaac, Emma, Ted, Thomas, Izzy from the UK, Iris from New Zealand, Julius from SIS Maennedorf in Switzerland, Henry from USA .<br></br> <br></br> Year 6 pupils from St John Fisher Harrogate Magic maths Club</span><br></br> <br></br> We started off by thinking of all the possible ways of making the totals. This took a long time.<br></br> We thought that it would be best to make the biggest totals first, using the bigger numbers to make them.<br></br> 14 = 9+5, 13 = 6 + 7, 1 + 2 = 3, 4+3 = 7 and 8 + 0 = 8.<br></br> Some of us did it the other way round, making the smallest totals first, with the smallest numbers.<br></br> 1 + 2 = 3, 4 + 3 = 7, 8 + 0 = 8, 7 + 6 = 13 and 9 + 5 = 14.<br></br> We could also come with pairs randomly but its quicker to use a strategy.<br></br> 7 + 0 = 7, 5 + 3 = 8, 9 + 4 = 13, 6 + 8 = 14 and 1 + 2 = 3.<br></br> <br></br> <span class="editorial">Ieuan from Riverside School, Prague in the Czech Republic wrote;</span><br></br> <br></br> At first I was randomly picking numbers, and on my first attempt doing it I found a , 0+7=7 5+3=8 9+4 =13 6+8=14 2+1=3. And then I tried using a system from then on off adding a number to the smaller number then subtracting one from the smaller number, but it did not go very well because when I converted 5+3 to 4+4 I realised that you cannot do that.<br></br> Then I found out something quite clever that from one onward each 2 numbers have the same amount of possibilities, for example 2 and 3 have 2 possibilities, 4 and 5 have have 3, 6 and 7 have 4, 8 and 9 have 5 and it goes on forever! So I wrote down all the possibilities for 7,8,13,14,3.<br></br> Then I shortened it so if i use 13 as an example,13+0, 12+1, 10+3, 9+4, 8+5, 7+6. Then I would take off 13+0, 12+1 and 10+3 and do that for all the rest! So when I had all the possibilities I did 2 attempts without succeding then I got one and I started explaining it on here, but I realised I had found the same as my first attempt. Then I did one attempt and I found another 3+0, 8+6, 9+4, 7+1 and 2+5.<br></br> <br></br> <span class="editorial">Well this has received the largest number of solutions I have had to read. A big well done to you all and I'm sorry I could not show everyone's thinking that they did to get a solution.</span><br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Sealed Solution</h2> <br></br> <p>A set of ten cards, each showing one of the digits from $0$ to $9$, is divided up between five envelopes so that there are two cards in each envelope. The sum of the cards inside it is written on each envelope:</p> <p class="c1"><mdo:image alt="five envelopes with the numbers 7, 8, 13, 14 and 3 written on them (one number per envelope)" height="35" src="envelopes.gif" width="288"></mdo:image></p> <p class="c2">What numbers could be inside the $8$ envelope?</p> <p class="acknowledgement c2" style="font-style: italic;">Thank you to Alan Parr for allowing us to adapt one of his problems.</p> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=1177&part=index">Sealed Solution</a> offers the chance for children to work in a systematic way and is a great context in which to encourage them to explain and justify their reasoning.</div> <br></br> <h3>Possible approach</h3> <div>Begin by familiarising children with the context: Using digit cards $0 - 5$, put $0$ and $1$ in one envelope and write their total on the outside (or on a "post-it" note stuck to the envelope). Put $3$ and $5$ in another envelope, again writing their total on the envelope. Explain that the other two cards will go in the last envelope. What will the total be? How do they know? Try this again, this time putting $0$ and $5$ in one envelope and recording the total. But then put two cards in another envelope without showing them to the children but writing only the total on the outside of the envelope. Repeat this for the third envelope. (For example you could have $1$ and $3$ in the first and $2$ and $4$ in the second.) What numbers are in the two envelopes? How do they know? Try again, this time keeping $0$ and $5$ in the first envelope but suggest that you want to put the other cards in pairs into the envelopes, so that the totals on the other two are the same. What could you do? How do they know? At each stage, children can be working in pairs, perhaps using mini-whiteboards and digit cards to try out their ideas.</div> <br></br> <div>Continue like this, gradually building up the level of complexity, and each time focusing on children's clear reasoning. You could put the cards in the envelopes without showing any to the children, only writing the totals. Invite the children to say which numbers could be in the envelopes. Next, using four envelopes and cards $0 - 7$, write $5$, $11$, $8$ and $4$ on the envelopes. How could they put numbers in the envelopes for the totals to be right? The final challenge could then be to solve the problem as it stands, with five envelopes and ten digit cards.</div> <br></br> <div>You could leave this final problem up on the wall for children to contribute solutions to over a longer period of time.</div> <br></br> <h3>Key questions</h3> <div>Which envelope shall we try first? Why?</div> <div>What could be in this envelope?</div> <div>Are there any numbers which you know definitely <span style="font-style: italic;">aren't</span> in this envelope? Why?</div> <div>Are there any other solutions?</div> <br></br> <h3>Possible extension</h3> <div>Children could make up their own problem along these lines.</div> <br></br> <h3>Possible support</h3> <div>Having digit cards available for children to use will free up their thinking and will make it easier to try out different ideas without worrying about crossing out on paper.</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> What are the possible ways of making the numbers on the envelopes?<br></br> Which number has the fewest possible combinations? It might be worth starting from this envelope and looking at what could be in the others. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Solutions prior to 2013<br></br> Sam sent us this solution:<br></br> <br></br> The possible ways of making the numbers on the envelopes are:<br></br> <br></br> 3: 1+2 or 0+3<br></br> 14: 5+9 or 6+8<br></br> 7: 0+7 or 1+6 or 2+5 or 3+4<br></br> 8: 0+8 or 1+7 or 2+6 or 3+5<br></br> 13: 4+9 or 5+8 or 6+7.<br></br> <br></br> Let's try 3=1+2 and 14=5+9. Then we have to have 13=6+7, and 8=0+8 (and 7=3+4).<br></br> <br></br> Let's try 3=1+2 and 14=6+8. Then we must have 13=4+9, and 7=0+7, so 8=3+5.<br></br> <br></br> Let's try 3=0+3 and 14=5+9. Then we must have 13=6+7, and 7=1+6, and this isn't allowed, so this isn't possible.<br></br> <br></br> Let's try 3=0+3 and 14=6+8. Then we must have 8=1+7 (and 7=2+5).<br></br> <br></br> So the possible pairs that are in envelope 8 are 0 and 8, or 3 and 5, or 1 and 7.</mdoxml> 2 4 0 1 0 0 0 Sealed Solution Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes? Decision Mathematics and Combinatorics Combinations Using, Applying and Reasoning about Mathematics Working systematically Calculations and Numerical Methods Addition & subtraction Mathematics Tools Digit cards Admin Upper primary mapping document